Questions: The graph of f'', the second derivative of the continuous function f, is shown above on the interval [0,9]. On this interval f has only one critical point, which occurs at x=6. Which of the following statements is true about the function f on the interval [0,9]? A f has a relative minimum at x=6 but not an absolute minimum. (B) The absolute minimum of f is at x=6. C. f has a relative maximum at x=6 but not an absolute maximum. (D) The absolute maximum of f is at x=6.

The graph of f'', the second derivative of the continuous function f, is shown above on the interval [0,9]. On this interval f has only one critical point, which occurs at x=6. Which of the following statements is true about the function f on the interval [0,9]?

A f has a relative minimum at x=6 but not an absolute minimum.
(B) The absolute minimum of f is at x=6.
C. f has a relative maximum at x=6 but not an absolute maximum.
(D) The absolute maximum of f is at x=6.

Transcript text: 9 Mark for Review The graph of $f^{\prime \prime}$, the second derivative of the continuous function $f$, is shown above on the interval $[0,9]$. On this interval $f$ has only one critical point, which occurs at $x=6$. Which of the following statements is true about the function $f$ on the interval $[0,9]$ ? A $f$ has a relative minimum at $x=6$ but not an absolute minimum. (B) The absolute minimum of $f$ is at $x=6$. C. $f$ has a relative maximum at $x=6$ but not an absolute maximum. (D) The absolute maximum of $f$ is at $x=6$.

Solution

Solution Steps

Step 1: Identify the critical point

The problem states that the function $ f $ has only one critical point, which occurs at $ x = 6 $.

Step 2: Analyze the second derivative at the critical point

The graph provided is of $ f'' $, the second derivative of $ f $. To determine the nature of the critical point at $ x = 6 $, we need to look at the value of $ f''(6) $.

Step 3: Determine the concavity at the critical point

From the graph, $ f''(6) $ is negative, indicating that $ f $ is concave down at $ x = 6 $. This suggests that $ x = 6 $ is a point of local maximum for $ f $.